Honours Finance (Advanced Topics in Finance: Nonlinear Analysis) Lecture 6: Dual Price Level Hypothesis continued Introduction to Chaos Government and Reflation Real model with government confirms Minsky on Big Government Anti-cyclical spending and taxation of government enables debts to be repaid Renewal of cycle once debt levels reduced What about the full Monty?: Cyclical economy Debt Prices Government A more complex picture still... Price dynamics One essential aspect of Keyness analysis: wage bargain set in nominal wage Easily introduced into Goodwin model: Phillips curve is now with respect to money wage: 1 dW Ph W dt W Pc w
W w L Pc w L w Y Yr Pc L a a Income distribution dynamic now includes inflation: d d w 1 d w 1d w a dt dt a a dt a a dt d 1 d W dt a dt Pc d 1 1 W Ph w p dt a Pc d 1 1 d W 1 d W
Pc dt a Pc dt Pc Pc dt d Ph p dt Workers share of output will grow if wage demands exceed the sum of inflation and productivity growth Price dynamics Next issue how to model prices Price equals marginal cost logic is out! Simplest alternative: Kaleckian markup pricing: W L Pc 1 Yr But other aspects of reality worth introducing Time lags in price setting (Kydland & Prescott, Blinder et al., estimate at 1 year) Counter-cyclical markups (unexpected Kydland & Prescott finding) Covering time lags: Same logic as shown in Advanced Political Economy lectures on modelling Circuitist theory: Exponential decay form with length of lag shown as inverse: Price dynamics Time lag p indicates how long before prices react
Expression in brackets gives level once rate of 1 W L 1 zero: change of prices equals d P P dt p Yr In Goodwin-Minsky system, can be reduced to d P dt 1 p 1 W L
YrP P 1 d P dt 1 p So inflation p now is:p P 1 1 1 w a d P
P dt 1 p d P dt 1 p P 1 1 1 1 Government and Reflation with Prices A still more complex six dimensional model: Model basically replicates the post-WWII economic record No Depression, but cycles, inflationary surges growing government deficit
Employment and Wages Share Complex aperiodic cycles Period 10-20 years, vs 4 in non-price model 1.027967 1.1 Proportion 1 < 1> Z i 0.9 < 2> Z i 0.8 0.7 0.618378 0.6 0
50 0 100 150 200 250 < 0> Z i Years Wages Share Employment 300 350 400 450 500 500 Debt Ratio and Price Level
Inflation counteracts tendency to accumulate debt Debt to Output Ratio 6 4 2 0 0 50 100 150 200 250 300 350 400 450
500 Years Price Level 2.5 Price Index 2 1.5 1 0.5 0 50 100 150 200 250 Years
300 350 400 450 500 Deflation avoided by countercyclic al government, but... Net Government Government runs an increasing deficit to restrain tendencies towards debt-deflation (assuming governments maintain a Keynesian approach!) Government Surplus to Output Ratio 8.89002 . 10 3 0.02
0 < 5> Z i < 6> Z i 0.02 0.04 0.056036 0.06 0 0 50 100 150 200 250
< 0> Z i Years 300 350 400 450 500 500 From Chaotic Limit Cycle to Strange Attractor 1.027967 1.05 1 Employment 0.95 < 2> Z i
0.9 0.85 0.8 0.76609 0.75 0.6 0.618378 0.62 0.64 0.66 0.68 0.7 0.72 < 1> Z i Wages Share 0.74
0.76 0.78 0.8 0.82 0.84 0.829551 From Chaotic Limit Cycle to Strange Attractor 2.000133 2.2 2 1.8 Price Level 1.6 1.4 < 4> Z i 1.2
1 0.8 0.6 0.547244 0.4 0.75 0.76609 0.8 0.85 0.9 < 2> Z i Employment 0.95 1 1.05 1.027967 From Chaotic Limit Cycle to Strange Attractor
5 4 3 2 0.8 0.7 < 1> < 2> < 3> Z ,Z ,Z 0.8 0.9 1 Falling debt ratio Rising price level And rising debt ratio Deficit rises to counterbalance Debt to Output Ratio
Rising wages share Net Government Surplus Increasing employment Wages Share of Output System Dynamics... 1.10 0.95 0.80 Falling employment Rising employment Wages Share of Output Employment Rate 0.65 450 460 470 6.0 Year
480 490 500 480 490 500 Debt to Output Ratio Commodity Price Index 4.5 3.0 1.5 0.0 450 460 0.01 470 Year Net Government Surplus
Deficit falls to counterbalance -0.02 -0.05 450 460 470 Year 480 490 500 Conclusion Dual Price Level Hypothesis Explains 19th century trade cycle and frequent Depressions 20th century avoidance of Depressions but financedriven cycles and countervailing government deficits Counters conventional analysis Finance (via debt) has macroeconomic impact Government countercyclical policies needed to counter tendency to debt-deflation but system still highly unstable
Inflation may contribute to avoidance of Depression Interest-rate based stabilisation policy may destabilise via debt amplification effects Complex behaviour from simple rules Complex behaviour of model contradicts conventional economics/finance beliefs about modelling Aperiodic cycles can be explained by deterministic system Long run of model need not be equilibrium Model exhibits complexity (modern alternative term for chaotic) Some basics on complexity... And three means chaos Original Goodwin two dimensional system had peculiar characteristic: Conservation law we start with d 1 dt v d w dt
Work out d/d: 1 d v d w Put in separable form: 1 w d v d And three means chaos Now integrate: 1
w d v d C Constant difference between two integrals conserved by system In practice means system constrained to a closed loop in (, space or phase plane Special case of general property of 2-dimensional systems No time paths of solution to deterministic ODE/system of ODEs can intersect Systems either diverge or converge to/from point or limit cycle. One of most complex examples: Van der Pols equation: Nonlinearity (but not Chaos) Van der Pols equation d 2I dI 2
I 1 I 0 2 dt dt explains cycles in electric current in vacuum tubes which the linearised equations d 2I dI 1 L 2 R I 0 dt dt C 2.012349 3 2
1 <1 > Z i <1 > X i 0 1 cant explain: 2 2.240069 3 4 3.540226 3 2 1 0 <2 >
<2 > Z , X i i 1 2 3 2.690631 And three means chaos But behaviour of trajectories constrained by 2dimensional, horizontal plane on which dynamics occur If trajectories cant intersect, then most complex behaviour which can occur is limit cycle However with three or more dimensions, trajectory in 3+D space can be infinitely complex chaotic long-term solutions, periodic orbits (limit cycles in 3+D), point attractors (equilibria), divergence For discontinuous systems (difference equations), chaos can occur even in 1D for non-invertible maps All high order ODEs and some second order can be converted into systems of 3+ first order ODEs Brief introduction to Chaos/Complexity An infinitely complex field (though not the same as Complex Analysis!: this is the mathematics of complex
numbers [a+ib]) see Gleik 87, Ott 93, Rosser 91, Lorenz 93, Brock 93 Essential characteristics Analytically insoluble differential/difference equations Though many theorems about behaviour of solutions proven Analytic techniques applied to equilibria, global stability etc. Sensitive dependence on initial conditions and apparently random behaviour accuracy of prediction from current conditions degrades rapidly Far from equilibrium dynamic behaviour in long run Brief introduction to Chaos/Complexity Two broad classes of Chaotic/Complex systems Conservative (or Hamiltonian) Some physical attribute conserved in some sense System can be characterised geometrically as space-distorting while still maintaining the volume enclosed by a space Many physical process models conservative Dissipative (or non-Hamiltonian) No physical attribute conserved Geometric analogy no longer possible Most biological/economic models dissipative Dissipative models (unlike conservative ones) have
attractors which can be stable/unstable, points/regions of space, etc. Brief introduction to Chaos/Complexity Much mathematical analysis of complexity done by converting continuous time systems (like DPL model) into discrete time by sampling at regular intervals mapping onto 2D surface Analysis of chaotic time series often done by mapping time series onto itself with a time lag (embedding) Time series generated by dissipative chaotic process has fractal (non-integer) dimension: Fractal Dimension Dimensionality a crucial aspect of an entity How many dimensions do you occupy? (If you answer three, then youre saying you are dead... and not decaying) Most mathematical abstractions have integer dimensions Point: 0 Line: 1 Square: 2, Cube: 3, etc. How well do such abstractions describe real-world objects?
Fractal Dimension Path followed by an ant Dimension 1, a line? Hardly: at best numerous line segments Dimension 2, a square? No: doesnt traverse every point in region between origin, end and sidedeviations A mountain Dimension 3, a cube? No: at best the sum of numerous mini-cubes Natural objects have fractional dimensions ants path between 1 and 2 mountain between 2 and 3 Fractal Dimension Dimensions of pure mathematical abstractions easy Dimensions of real world objects requires a concept of measurement what is the minimum size of something needed to completely describe the coverage of an object? One method: the box-counting dimension (Ott: 6971) How many squares of length does it take to completely cover an object as Formula is ln N D lim 0 ln 1 Consider for: 2 points; line; solid shape (standard
mathematical abstractions) Fractal Dimension For two points ln 2 D lim 0 1 0 ln For a line (length l) D lim ln 1 ln l 0
ln l ln 1 lim ln 1 0 ln l ln 1 lim ln 1 0 ln 1 lim 0 ln 1 1 Relatively tiny number as 0 compared to 1/ Fractal Dimension For a closed curve (Area A): D lim 0
Relatively tiny number as compared to 1/ How about a non-integer object? Example: the Cantor set Take line length 1 and remove middle third repeat process with each new line so created How many (1-dimensional) boxes needed to cover it? Fractal Dimension Cantor Set: At each stage (n) of construction box of length (1/3)n needed to contain each segment 2n boxes needed Fractal Dimension Working this out: ln N 0 ln 1 ln 2 n lim
n ln 1 n 1 3 n ln 2 lim ln 3n n n ln 2 lim n n ln 3 ln 2 0.63 ln 3 D lim
So Cantor Set has fractional (Fractal) dimension Similar analysis applied to Finance data if generated by deterministic (as well as stochastic) processes then will have fractional dimension Nonlinear analysis of complex systems To date weve shown trade cycle can be modelled as nonlinear dynamical system nonlinear dynamical systems can generate time series which appear random actual time series will have both deterministic and stochastic processes behind them It can be argued therefore that financial time series are (in part) driven by deterministic nonlinear (in part) self-referential processes Can we (approximately) recreate the nonlinear process behind the time series, and thus (for a short while) predict its future course?
Nonlinear analysis of complex systems Yes (sort of); but very difficult to distinguish deterministic chaos from random noise reconstruction not a proper model as in DPL or even regression equation as in (linear) econometrics Uncovering determinism in randomness Recapping trade cycle can be modelled as nonlinear dynamical system nonlinear dynamical systems can generate time series which appear random actual time series will have both deterministic and stochastic processes behind them It can be argued therefore that financial time series are (in part) driven by deterministic nonlinear (in part) self-referential processes Can we (approximately) recreate the nonlinear process behind the time series, and thus (for a short while) predict its future course? Uncovering determinism in randomness Yes, with reservations difficulty of distinguishing randomness from chaos random numbers generated by computers are in fact produced using deterministic programs same seed number generates identical stream
of numbers real data from any given dynamic system an overlay of endogenous (mainly deterministic) and exogenous (stochastic) forces any social system will have input of human decision-making rapid loss of information in any complex/chaotic system (over to logistic_error.mcd)... Techniques to uncover structure Two key methods in vogue (but also infancy): Neural networks Genetic Algorithms Neural networks based on analogy to structure and (presumed) function and methodology of brain Human brain has about 100,000,000,000 neurones Each neurone has about 1000 connections with other neurones Inputs from 1000 input neurones determine whether and how much a neurone will fire Brain a complex network of neural structures... Neural Networks Computer analogy to brains neural network incredibly simplistic by comparison: one set of inputs, one input per piece of sensory data one hidden layer of neurones, the brain of network
one layer of outputs, one per relevant output (normally) each node in each layer connected to every node in next layer connection via nonlinear weighting functions (typically logistic curve) where weights (values of parameters for function) alterable by network Neural Networks Input Layer Hidden Layer(s) Output Layer O U T P U T S I N P U T S
Weight Function Neural Networks Neural network trained on historical data (for time series prediction), object images (for recognition [e.g.] OCR), etc. Training alters parameter values of weights functions for all inputs Training continues until weights stabilise, predictions of network approximate to actual data Network then used in practice Predict next value of Dow Jones Recognise previously unseen fonts/handwriting Control movements of robotic welding arm Neural network behaviour Basic pattern is Network trained on existing data Shown combination of input data and desired output (numerically coded if necessary) Initially random parameters on weights functions generate incorrect outputs Errors used (in modified least squares manner) to alter weights Process continues until errors minimise/stabilise Network used to analyse new data The importance of being nonlinear
Earliest form of neural net was perceptron Same basic architecture as today, but used linear weights functions Initial enthusiasm for concept waned when shown that perceptron could not solve exclusive OR (XOR) problem: Given two binary inputs, produce an output of 1 when one but not both of the inputs are also 1 In truth table form: XOR Case 1 Case 2 Case 3 Case 4 Input A Input B Output 0 1 0 1 0 0 1 1 0 1 1
0 The importance of being nonlinear Basic structure of perceptron is Output Inputs (1 or 0) are multiplied by weights to produce output Weights adjusted using least squares until output clearly distinguishes between input patterns Weights adjustment effectively moves location of straight line on a plane Consider example of AND: w1*x1 Node 1 w2*x2 Node 2 x1 Input 1 x2
Input 2 The importance of being nonlinear Truth table is: AND Input A Input B Output Case 1 0 0 0 Case 2 1 0 0 Case 3 0 1 0 Case 4 values 1 for parameters 1 1 Perceptron problem is to find
for a line such that cases 1 to 3 can be distinguished from case 4. Straight line equation is a x b y c Can we find values for a and b so that, for an arbitrary c, cases 1 to 3 lie on one side of the line, and case 4 on the other? No problem: The importance of being nonlinear Input 2 Many lines will give valid result This side produces output of 1 (0,1) (1,1) This side produces output of 0 (0,0) (1,0) Input 1
The importance of being nonlinear Lets look at XOR problem the same way: Input 2 (0,1) This side produces output of 1 (1,1) This side produces output of 0 (0,0) (1,0) No matter where you draw the line, you cant get both red dots on one side and both green dots on the other Input 1 The importance of being nonlinear Input 2 This side produces output of 0 (0,1)
This side produces output of 1 (0,0) Nonlinear shape (1,1) can easily distinguish the inputs correctly (1,0) Input 1 Genetic Programming Based on an analogy to evolution Evolution has produced most complex problem solving systems of all using simple operations crossover (sexual reproduction at DNA level) mutation under pressure of environmental selection where environment is in large measure a product of evolution itself (e.g., without impact of carbon dioxide to oxygen-fixing bacteria, Earths atmosphere would be mainly carbon dioxide) Genetic Programming Computer analogy of evolution produces population of randomly varying programming fragments (If, While, And, Or etc.)
Assesses ability of each program to replicate key feature of data e.g., take economic data and reproduce change in exchange rate Least Fit programs deleted Fittest programs reproduced with codeswapping (crossover) and random alteration (mutation) System run for many generations to select best program(s) Genetic Programming Hypothetical program fragment: Or And And NOT D0 D1 D0 D1 Many such fragments
tested against data Those with better fit selected for breeding crossover of one half of one program with another random alteration to program Runs continue till fits to data stabilise
Grouping Two methods of grouping: Ungrouped data can be "grouped" during analysis Grouping in the field at the time of the survey Grouping after the survey Done to improve model fit After detecting lack of model fit Due to heaping...
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